Concepts of Infinity

0.999... = 1


You can get a one-to-one correspondence from the interval [0,1] to [0,10] by the map f with

f: x -> 10*x

and [0,1] and [0,10] has the same cardinality.

Mathematically Infinitity isn't a number - its a concept. It has some properties tha are fixed and some that are variable.

In maths you can size and infinity, by this I mean you compare two groups and say one infinity is 'bigger' than another. This does not mean that is has more elements (which it kinda does) but is strictly understood to mean that as expressed as an infinite series it grows faster than another infinite series.

For instance if G = the sum of all integers from 1 .. n as n approaches infinity and H = the sum of all integers doubled from 1..n etc... then G grows slower than H and H is 'bigger' than G.

Now 0.9 repeater, = 3/3 as 1/3 = 0.3 repeater and 3 * 1/3 = 3 * 0.3 repeater, and 3/3 = 1.

So 99.9 repeater is by defintion 100.

Hope that helps. I am not sure of the precise, correct technical terms to use when comparing the relative size of two sets. My tutor (PhD candidate - now a senior lecturer) used to say 'bigger'.

I can give two simple answers. One the first question, we know that three thirds equals one. One third is expressed .3333..., and two thirds is .66666...., and by the same math, three thirds is .99999.... which would have to equal one.

On the second question, infinity is the same size, but is restricted by possibilities in one example more than the other. Infite possibilities does not mean ALL possibilities. If there were infinite stars in the universe, that would not mean that one of them could talk and juggle planets, however if such were the case, then you would have more diversity among the stars, but it has nothing to do with the fact that there are infinite stars. (emphasis on every "If")

I agree with satt_focusable's answer to your second question. I just wanted to add one more example of two infinite sets that have the same cardinality. Take the set of all the positive integers (1, 2, 3, ...) and then take the set of just the even ones (2, 4, 6, ...), it seems that the second set would have half as many members as the first set since it is made up of every other number from the first set. The interesting thing about it is that both sets have exactly the same number of elements. I think this idea has even been used as a definition of 'infinity': that a set has an infinite number of elements if and only if there is a subset of that set that has the same number of elements as the original set. So looking back at your original question, we know that the set of [real] numbers between 0 and 10 is infinite BECAUSE it has the same number of elements as the set of [real] numbers between 0 and 1.

On a second look at your question I realized I really didn't answer it. you asked whether one infinite set can have more members than another infinite set. The answer is 'yes.' This one will be harder to explain. Lets look at three different sets:

PI- the set of positive integers: 1, 2, 3, ...
RAT- the set of rational numbers between 0 and 10
REAL- the set of real numbers between 0 and 10

RAT and PI have the same infinite number of members, but REAL has a different (and bigger) infinite number of members. To see this we have to use the idea of one-to-one correspondence. this process is pretty easy when one of the sets is PI since all you really have to do is count the members of the set that you want to correspond PI to. You can't really count all the members since we're dealing with infinite sets, but if you devise a method of counting that includes ALL the members of the set then we can say that PI and the set we are counting have a one-to-one correspondence. So lets devise a way to count RAT. We'll start with the whole numbers 0, 1, 2, ..., 9, 10. Then we'll move on to the first decimal place (leaving out the whole numbers) .1, .2, .3, ..., 9.8, 9.9. After that the second decimal place .01, .02, .03, ..., 9.98, 9.99. If we keep on moving over decimal places we will, in the end, count all of RAT, therefore RAT and PI have the same infinite number of members.

There are, however, many irrational numbers that we haven't counted, like pi and the sqare root of 2. How do we know we haven't counted them? Because every number that has been counted has had a finite decimal expansion (even those near the end of the counting procedure) though pi and sqrt2 both have infinite non-repeating decimal expansions. Moreover, there is no way to count all the numbers between 0 and 10 that have infinite non-repeating decimal expansions (If you can think of one, please let me know- I would be really interested). So we have found two infinite sets that cannot fit into one-to-one correspondence with each other, which leads us to the conclusion that some infinities have 'more' members than others, i.e. some infinite sets are of a higher cardinality than other infinite sets.

Hopefully this helped more than it confused

One infinity cannot be a higher number than another infinity. For reasons which G_day covers. If an infinite set includes numbers which another infinite set does not, then it is merely a more diverse infinity.

The set consisting of 0,1,2,.. is smaller than the set of real numbers x such that 0<x<1, in the cardinality.

I like the Hitch Hiker's Guide to the Galaxy's definition of Infinite, the old "You think its a long way to the cornershop? Well space is much, much bigger...

OOPS, I thought it said "concepts of infedelity"....lol

I recall reading about a branch of mathematics that dealt with Transinfinite Sets.
These were numbers that were bigger than any other infinity.

You can size infinities - but don't call them numbers!

Tarski's paradox.

Wasn't Tarski's paradox you could say a contradictory, self-referential expression in any sophisticated language. If you assumed it resolved to a final state you reached a contradiction - unless you exotically warped definitions like to say it wasn't a valid sentence in the first place. Note the ability to do these twists occur in any powerful enough logic system or language - be it maths or English?

Basically you say this sentence is false or even better this sentence is not true. Most folks say Logic dictates the sentence must be either true or false, but think carefully about this major assumption for thinking this way creates a cute paradox meaning you have to play with definitions to treat it. I view these types of expressions as having an inconsistent unresolved forms and don't resolve - they way a qubit has multiple states before being resolved.

In electronic terms - this expression doesn't simply describe a logic circuit - rather its its a logic circuit with a memory loop that has inverted feedback and an unresolved initial state and you are asked to make a determination as to its final state as if it actually does settles into a final state. But such a circuit does not converge to a final state, so assuming it does sets yourself up for a fall.

A trivial example where if you misconstrue how to frame your analysis you confuse many folk!

Hmm... how about this question. What if you take the infinite set of all numbers from zero up, and subtract the infinite set of even all even numbers from zero up? That's more interesting to me than the initial question.

The sets of all the natural numbers, all the even numbers, and all the odd numbers have the same cardinality.
The set of real numbers has different cardinality from the set of natural numbers.

But that's what I find interesting. If you take the set of all natural numbers, and subtract (not sure if that's the best word) the set of all even numbers the set becomes much less diverse, despite its cardinality. The cardinality remains the same, even though it was reduced by an infinite set?

SCoates - you simply get the infinite set of all odd positive integers? Still an infinite cardinality by the definition:

The cardinality of a set is the number of elements in the set. For finite sets this is always a natural number.
Countable infinite sets have cardinality ℵ0 ('aleph-null'); the set of real numbers has cardinality ℵ1.
The cardinality of a set A can be written n(A), |A|, or A

But that's what I find interesting. The set has been reduced by an infinite number of items, and still retains the same cardinality.

It is one of ways to define the infinite set.


Even though an APPARENT size of information contained in a set is reduced, the cardinality of the set can be the same, and the seemingly omitted information can be retrieved through a one-to-one correspondence.